- Research Article
- Open Access

# Some Fixed Point Theorems on Ordered Metric Spaces and Application

- Ishak Altun
^{1}Email author and - Hakan Simsek
^{1}

**2010**:621469

https://doi.org/10.1155/2010/621469

© I. Altun and H. Simsek. 2010

**Received:**2 July 2009**Accepted:**13 January 2010**Published:**21 January 2010

## Abstract

We present some fixed point results for nondecreasing and weakly increasing operators in a partially ordered metric space using implicit relations. Also we give an existence theorem for common solution of two integral equations.

## Keywords

- Continuous Function
- Integral Equation
- Point Theorem
- Existence Theorem
- Contractive Condition

## 1. Introduction

Existence of fixed points in partially ordered sets has been considered recently in [1], and some generalizations of the result of [1] are given in [2–6]. Also, in [1] some applications to matrix equations are presented, in [3, 4] some applications to periodic boundary value problem and to some particular problems are, respectively, given. Later, in [6] O'Regan and Petruşel gave some existence results for Fredholm and Volterra type integral equations. In some of the above works, the fixed point results are given for nondecreasing mappings.

We can order the purposes of the paper as follows.

First, we give a slight generalization of some of the results of the above papers using an implicit relation in the following way.

In [1, 3], the authors used the following contractive condition in their result, there exists such that

Afterwards, in [2], the authors used the nonlinear contractive condition, that is,

where is anondecreasing function with for , instead of (1.1). Also in [2], the authors proved a fixed point theorem using generalized nonlinear contractive condition, that is,

for , where is as above. In the Section 3, we generalized the above contractive conditions using the implicit relation technique in such a way that

for , where is a function as given in Section 2. We can obtain various contractive conditions from (1.4). For example, if we choose

in (1.4), then, we have (1.3). Similarly we can have the contractive conditions in [7–9] from (1.4).

In some of the above mentioned theorems, the fixed point results are given for nondecreasing mappings. Also in these theorems the following condition is used:

In Section 4, we give some examples such that two weakly increasing mappings need not be nondecreasing. Therefore, we give a common fixed point theorem for two weakly increasing operators in partially ordered metric spaces using implicit relation technique. Also we did not use the condition (1.6) in this theorem. At the end, to see the applicability of our result, we give an existence theorem for common solution of two integral equations using a result of the Section 4.

## 2. Implicit Relation

Implicit relations on metric spaces have been used in many articles. See for examples, [10–15].

Let denote the nonnegative real numbers, and let be the set of all continuous functions satisfying the following conditions:

is nonincreasing in variables ;

Example 2.1.

Let and . If then which implies , a contradiction. Thus and . Similarly, let and then If , then Thus is satisfied with . Also , for all . Therefore, .

Example 2.2.

Let and . If then which is a contradiction. Thus and Similarly, let and then we have If then Thus is satisfied with Also for all Therefore, .

Example 2.3.

where is right continuous and , for

Let and If then which is a contradiction. Thus and Similarly, let and then we have If , then Thus is satisfied with Also , for all Therefore, .

Example 2.4.

Let and Then Similarly, let and then we have If then Thus is satisfied with . Also , for all . Therefore, .

## 3. Fixed Point Theorem for Nondecreasing Mappings

We need the following lemma for the proof of our theorems.

Lemma 3.1 (See [16]).

Let be a right continuous function such that for every , then , where denotes the -times repeated composition of with itself.

Theorem 3.2.

hold. If there exists an with , then has a fixed point.

Proof.

From , we have . Therefore, letting in (3.16), we have This is a contradiction since for Thus is a Cauchy sequence in so there exists an with .

which is a contradiction to . Thus and so .

Remark 3.3.

Note that if we take that

Instead of in Theorem 3.2, again we can have the same result.

If we combine Theorem 3.2 with Example 2.1, we obtain the following result.

Corollary 3.4.

hold. If there exists an with , then has a fixed point.

Remark 3.5.

Theorem of [2] follows from Example 2.3, Remark 3.3, and Theorem 3.2.

Remark 3.6.

We can have some new results from other examples and Theorem 3.2.

Remark 3.7.

and hence has a unique fixed point. If condition (3.27) fails, it is possible to find examples of functions with more than one fixed point. There exist some examples to illustrate this fact in [3].

## 4. Fixed Point Theorem for Weakly Increasing Mappings

Now we give a fixed point theorem for two weakly increasing mappings in ordered metric spaces using an implicit relation. Before this, we will define an implicit relation for the contractive condition of the theorem.

Let be the set of all continuous functions satisfying and the following conditions:

We can easily show that, all functions in the Examples in Section 2 are in .

Definition 4.1 (See [17, 18]).

Let be a partially ordered set. Two mappings are said to be weakly increasing if and for all .

Note that, two weakly increasing mappings need not be nondecreasing.

Example 4.2.

then it is obvious that and for all . Thus and are weakly increasing mappings. Note that both and are not nondecreasing.

Example 4.3.

Let be endowed with the coordinate ordering, that is, and . Let be defined by and , then and . Thus and are weakly increasing mappings.

Example 4.4.

Thus and are weakly increasing mappings. Note that but , then is not nondecreasing. Similarly, is not nondecreasing.

Theorem 4.5.

hold, then and have a common fixed point.

Remark 4.6.

Note that, in this theorem we remove the condition "there exists an with '' of Theorem 3.2. Again we can consider the result of Remark 3.7 for this theorem.

Proof of Theorem 4.5.

From , we have . Therefore, letting in (4.32), we have . This is a contradiction since for . Thus is a Cauchy sequence in , so is a Cauchy sequence. Therefore, there exists an with .

which is a contradiction to . Thus and so .

Remark 4.7.

We can have some new results from Theorem 4.5 with some examples for .

For example, we can have the following corollary.

Corollary 4.8.

hold, then and have a common fixed point.

Proof.

Let , then it is obvious that . Therefore, the proof is complete from Theorem 4.5.

## 5. Application

Consider the integral equations

The purpose of this section is to give an existence theorem for common solution of (5.1) using Corollary 4.8. This section is related to those [19–22].

Let be a partial order relation on .

Theorem 5.1.

Consider the integral equations (5.1).

Then the integral equations (5.1) have a unique common solution in .

Proof.

Then is a partially ordered set. Also is a complete metric space. Moreover, for any increasing sequence in converging to , we have for any . Also for every , there exists which is comparable to and [6].

Hence for each comparable . Therefore, all conditions of Corollary 4.8 are satisfied. Thus the conclusion follows.

## Declarations

### Acknowledgment

The authors thank the referees for their appreciation, valuable comments, and suggestions.

## Authors’ Affiliations

## References

- Ran ACM, Reurings MCB:
**A fixed point theorem in partially ordered sets and some applications to matrix equations.***Proceedings of the American Mathematical Society*2004,**132**(5):1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, El-Gebeily MA, O'Regan D:
**Generalized contractions in partially ordered metric spaces.***Applicable Analysis*2008,**87**(1):109–116. 10.1080/00036810701556151MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, Rodríguez-López R:
**Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations.***Order*2005,**22**(3):223–239. 10.1007/s11083-005-9018-5MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, Rodríguez-López R:
**Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations.***Acta Mathematica Sinica*2007,**23**(12):2205–2212. 10.1007/s10114-005-0769-0MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, Pouso RL, Rodríguez-López R:
**Fixed point theorems in ordered abstract spaces.***Proceedings of the American Mathematical Society*2007,**135**(8):2505–2517. 10.1090/S0002-9939-07-08729-1MathSciNetView ArticleMATHGoogle Scholar - O'Regan D, Petruşel A:
**Fixed point theorems for generalized contractions in ordered metric spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(2):1241–1252. 10.1016/j.jmaa.2007.11.026MathSciNetView ArticleMATHGoogle Scholar - Ćirić LjB:
**Generalized contractions and fixed-point theorems.***Publications Institut Mathématique*1971,**12**(26):19–26.MathSciNetMATHGoogle Scholar - Ćirić LjB:
**Fixed points of weakly contraction mappings.***Publications Institut Mathématique*1976,**20(34):**79–84.MathSciNetMATHGoogle Scholar - Ćirić LjB:
**Common fixed points of nonlinear contractions.***Acta Mathematica Hungarica*1998,**80**(1–2):31–38.MathSciNetMATHGoogle Scholar - Altun I, Turkoglu D:
**Some fixed point theorems for weakly compatible multivalued mappings satisfying an implicit relation.***Filomat*2008,**22**(1):13–21. 10.2298/FIL0801011AMathSciNetView ArticleMATHGoogle Scholar - Altun I, Turkoglu D:
**Some fixed point theorems for weakly compatible mappings satisfying an implicit relation.***Taiwanese Journal of Mathematics*2009,**13**(4):1291–1304.MathSciNetMATHGoogle Scholar - Imdad M, Kumar S, Khan MS:
**Remarks on some fixed point theorems satisfying implicit relations.***Radovi Matematički*2002,**11**(1):135–143.MathSciNetMATHGoogle Scholar - Popa V:
**A general coincidence theorem for compatible multivalued mappings satisfying an implicit relation.***Demonstratio Mathematica*2000,**33**(1):159–164.MathSciNetMATHGoogle Scholar - Sharma S, Deshpande B:
**On compatible mappings satisfying an implicit relation in common fixed point consideration.***Tamkang Journal of Mathematics*2002,**33**(3):245–252.MathSciNetMATHGoogle Scholar - Turkoglu D, Altun I:
**A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying an implicit relation.***Sociedad Matemática Mexicana. Boletín*2007,**13**(1):195–205.MathSciNetMATHGoogle Scholar - Matkowski J:
**Fixed point theorems for mappings with a contractive iterate at a point.***Proceedings of the American Mathematical Society*1977,**62**(2):344–348. 10.1090/S0002-9939-1977-0436113-5MathSciNetView ArticleMATHGoogle Scholar - Dhage BC:
**Condensing mappings and applications to existence theorems for common solution of differential equations.***Bulletin of the Korean Mathematical Society*1999,**36**(3):565–578.MathSciNetMATHGoogle Scholar - Dhage BC, O'Regan D, Agarwal RP:
**Common fixed point theorems for a pair of countably condensing mappings in ordered Banach spaces.***Journal of Applied Mathematics and Stochastic Analysis*2003,**16**(3):243–248. 10.1155/S1048953303000182MathSciNetView ArticleMATHGoogle Scholar - Ahmad B, Nieto JJ:
**The monotone iterative technique for three-point second-order integrodifferential boundary value problems with****-Laplacian.***Boundary Value Problems*2007,**2007:**-9.Google Scholar - Cabada A, Nieto JJ:
**Fixed points and approximate solutions for nonlinear operator equations.***Journal of Computational and Applied Mathematics*2000,**113**(1–2):17–25. 10.1016/S0377-0427(99)00240-XMathSciNetView ArticleMATHGoogle Scholar - Ćirić LjB, Cakić N, Rajović M, Ume JS:
**Monotone generalized nonlinear contractions in partially ordered metric spaces.***Fixed Point Theory and Applications*2008,**2008:**-11.Google Scholar - Nieto JJ:
**An abstract monotone iterative technique.***Nonlinear Analysis: Theory, Methods & Applications*1997,**28**(12):1923–1933. 10.1016/S0362-546X(97)89710-6MathSciNetView ArticleMATHGoogle Scholar

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